On one-point intersection property for self-similar fractals

This was a long-standing question since 90s whether one-point intersection property for a self-similar set implies open set condition (OSC). We answer this question negatively. We give an example of a totally disconnected self-similar set K subset of R which does not have OSC and has minimal overlap of its pieces, that is, all intersections of its pieces K-i boolean AND k(j), i not equal j are empty except only one, which is a single point.